Converse Ackermann property and constructive negation defined with a negation connective

Abstract

The Converse Ackermann Property is the unprovability of formulas of the form (A → B) → C when C does contain neither → nor ¬. Intuitively, the CAP amounts to rule out the derivability of pure non-necessitive propositions from non-necessitive ones. A constructive negation of the sort historically defined by, e.g., Johansson is added to positive logics with the CAP in the spectrum delimited by Ticket Entailment and Dummett's logic LC.